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I am looking for an analytic solution for the following two equations in the variables $v(x,t)$ and $u(x,t)$: $$ \begin{cases} \dfrac{\partial v}{\partial x} = -m\dfrac{\partial u}{\partial t} \\ \dfrac{\partial u}{\partial x} =-n \dfrac{\partial v}{\partial t} -av^5 \end{cases} $$ The boundary conditions are $$ v(0,t)=E\qquad u(l,t)=0 $$ The constants $m,n,a,E$ and $l$ are positive and non-zero. Thanks in advance.

Note: This is a simplified version of my earlier (unanswered) question posted on February 13, 2019.

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There's an obvious analytic answer:

$$ v(x,t) = E\qquad u(x,t) = a E^5 (l-x) $$

Is that what you wanted?

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  • $\begingroup$ Thanks very much. Apparently, there are more than one solution. This suggested answer is mathematically sound and interesting. As a matter of fact, I am looking for a solution(s) that also depend on the two other constants m and n. I think that the proposed answer is a special solution that prevails when t approaches infinity. $\endgroup$
    – user135784
    Commented Feb 24, 2019 at 17:32

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